On the connection problem for the second Painlevé equation with large initial data

Long, Wen-Gao; Zeng, Zhao-Yun

doi:10.48550/arxiv.2005.03440

Cited by 2 publications

(2 citation statements)

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“…Reference [3] investigates the applications of nonlinear eigenvalue problems to the first and second Painlevé equations and obtains the asymptotic behavior of their eigenvalues by relating these equations to the Schrödinger equation with PT -symmetric Hamiltonian H = p2 + gx 2 (ix) , with = 1 and 2, respectively. References [19,20] obtain the same results at a rigorous level for the first and second Painlevé equations, respectively. Remarkably, the large eigenvalues of the fourth Painlevé are also related to the eigenvalues of the PT -symmetric Hamiltonian with = 4 [4].…”

Section: Summary and Concluding Remarkssupporting

confidence: 55%

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Komijani¹

2021

J. Phys. A: Math. Theor.

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Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as ( 1 − 2 n ) / Γ ( r 2 n − 1 ) , where r k is the kth root of the digamma function.

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Section: Summary and Concluding Remarkssupporting

confidence: 55%

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Komijani¹

2021

J. Phys. A: Math. Theor.

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“…These laws were first found numerically [1,2]. They were then proved rigorously [3,4]. Curiously, these power laws are deeply connected to the linear eigenvalues of some non-Hermitian Hamiltonians in the WKB approximation [5].…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable nonlinear eigenvalue problems

Wang¹

2020

Preprint

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The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.

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On the connection problem for the second Painlevé equation with large initial data

Cited by 2 publications

References 20 publications

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Exactly solvable nonlinear eigenvalue problems

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